Question: Simplify and expand the following expression: $ \dfrac{3n + 8}{5n + 2}-\dfrac{2n + 1}{5n - 10} $
Solution: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(5n + 2)(5n - 10)$ Multiply the first term by $\dfrac{5n - 10}{5n - 10}$ $ \begin{align*} \dfrac{3n + 8}{5n + 2} \times \dfrac{5n - 10}{5n - 10} & = \dfrac{(3n + 8)(5n - 10)}{(5n + 2)(5n - 10)} \\ & = \dfrac{15n^2 + 10n - 80}{(5n + 2)(5n - 10)}\end{align*} $ Multiply the second term by $\dfrac{5n + 2}{5n + 2}$ $ \begin{align*} \dfrac{2n + 1}{5n - 10} \times \dfrac{5n + 2}{5n + 2} & = \dfrac{(2n + 1)(5n + 2)}{(5n - 10)(5n + 2)} \\ & = \dfrac{10n^2 + 9n + 2}{(5n - 10)(5n + 2)}\end{align*} $ Now we have: $ = \dfrac{15n^2 + 10n - 80}{(5n + 2)(5n - 10)} - \dfrac{10n^2 + 9n + 2}{(5n - 10)(5n + 2)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{15n^2 + 10n - 80 - (10n^2 + 9n + 2)}{(5n + 2)(5n - 10)} $ $ = \dfrac{15n^2 + 10n - 80 - 10n^2 - 9n - 2}{(5n + 2)(5n - 10)} $ $ = \dfrac{5n^2 + n - 82}{(5n + 2)(5n - 10)}$ Expand the denominator: $ = \dfrac{5n^2 + n - 82}{25n^2 - 40n - 20}$